Explicit universal sampling sets in finite vector spaces

TL;DR

This paper develops explicit sampling sets and efficient reconstruction algorithms for Fourier signals in finite vector spaces, enabling accurate approximation with a focus on signals with sparse Fourier coefficients.

Contribution

It introduces explicit sampling sets and algorithms tailored for Fourier signals in finite vector spaces, with complexity bounds and approximation guarantees.

Findings

Sampling sets of size O(pt^2r^2) and O(pt^2r^3 log p) constructed

Algorithms achieve near-optimal approximation with t non-zero Fourier coefficients

Fastest algorithm operates in O(p^2 t^2 r^3 log p) time

Abstract

In this paper we construct explicit sampling sets and present reconstruction algorithms for Fourier signals on finite vector spaces $G$, with $|G|=p^r$ for a suitable prime $p$. The two sets have sizes of order $O(pt^2r^2)$ and $O(pt^2r^3\log(p))$ respectively, where $t$ is the number of large coefficients in the Fourier transform. The algorithms approximate the function up to a small constant of the best possible approximation with $t$ non-zero Fourier coefficients. The fastest of the algorithms has complexity $O(p^2t^2r^3\log(p))$.